Software for performing meaningful analysis increasingly shows up on the desktops of design engineers.
A finiteelement model can be thought of as a system of solid springs. When a load is applied to the structure, all elements deform until all forces balance. For each element in the model, equations can be written relating displacements and forces at the nodes.
The element shown here, for example, is a 2D quadrilateral having four nodes. Each node has two degrees of freedom associated with it (displacements in X and Y directions), so that the element has a total of eight degrees of freedom. There must also be a nodal force for each nodal degree, so there are also eight nodal forces for the element.
These displacements and forces are identified by a coordinate numbering system for entry the computer program. For example, dxi1 is the deflection in the X direction for element i at node 1, while dyi1 is the deflection in the Y direction for the same node in the same element. Forces are identified in a similar manner, so that Fxi1 is the force in the X direction for element i at node 1.
An equation relating displacements and forces for the element takes the form of basic spring equation, F = kd. For four nodes:

k_{11}d_{xi1} + k_{12}d_{yi1} + k_{13}d_{xi2} + k_{14}d_{yi2} + . . . + k_{18}d_{yi4} = F_{xi1}

k_{21}d_{xi1} + k_{22}d_{yi1} + k_{23}d_{xi2} + k_{24}d_{yi2} + . . . + k_{28}d_{yi4} = F_{yi1}

k_{31}d_{xi1} + k_{32}d_{yi1} + k_{33}d_{xi2} + k_{34}d_{yi2} + . . . + k_{38}d_{yi4} = F_{xi2}

k_{81}d_{xi1} + k_{82}d_{yi1} + k_{83}d_{xi2} + k_{84}d_{yi2} + . . . + k_{88}d_{yi4} = F_{yi4}